On incomplete polynomials
For \(\ell \in {\mathbb{N}}\) let \(\bar n=(n_ 1,n_ 2,...,n_{\ell})\) be an \(\ell\)-tuple of integers \(n_{\nu}\) satisfying \(0\leq n_ 1<n_ 2<...<n_{\ell}\), and \(V_{\bar n}=span(x^{n_ 1},x^{n_ 2},...,x^{n_{\ell}})\), considered on the interval [a,b], where \(0\leq a<b<\infty\). The paper is concerned with the best \(L_ p\)-approximation (1\(\leq p\leq \infty)\) of polynomials \(p\in P_ d\), i.e. of degree at most d, by functions from \(V_{\bar n}\). If S is a given set of nonnegative integers with \(\ell \leq card(S)\) then we denote by \(S_{\ell}\) the set of all those \(\ell\)-tuples \(\bar n\) for which \(n_{\nu}\in S\), \(\nu =1,2,...,\ell\), holds. Furthermore let \(U\{V_{\bar n}| \bar n\in S_ 1\}=\Pi_ 1(S)\). After referring to results in approximating monomials by functions from \(\Pi_ 1(S)\) the question is asked, if there exists always a number M, depending on S, d and on \(\ell\), such that any best approximating polynomial \(q\in \Pi_ 1(S)\) to any polynomial \(p\in P_ d\) satisfies degree \(q\leq M\). For \(p<\infty\) the answer is affirmative. The proof of this assertion will be published elsewhere. For \(p=2\) and \(\ell =1\) and for the interval [0,1] a simple proof is presented in this paper. Furthermore it is proved that for \(p=\infty\) and \(2\ell \leq d+1\), where \(a>0\) holds, the following assertion is valid: To every polynomial \(q\in \Pi_ 1(S)\) there exists a polynomial \(p\in P_ d\), such that q is the unique best approximation to p with respect to \(\Pi_ 1(S)\). - Hence the answer to the original question is in the negative. Two open problems closely connected with the main question are stated.
scientific article; zbMATH DE number 3901378